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# Induced current in a wire

Video transcript

Let's say I have a
magnetic field popping out of this video. So these little brown circles
show us the tips of the vectors popping out
of our screen. And in that magnetic field I
have this wire, this off-white colored wire. And sitting on that off-white
colored wire I have a charge of charge Q. Let me write down
the other stuff. So this is the magnetic
field B coming out. Let's say I were to take
this whole wire. And let's say that the wire
overlaps with the magnetic field a distance of L. So let's say the magnetic
field stops here and stops here. And let's say that this distance
right here, that distance is L. I drew it a little bit weird
but you get the idea. From here to here is L. I have this charge sitting on
some type of conductor that we can consider a wire. And the magnetic field is
pushing out of the page. So in this current formation,
let's say I don't have any voltage across this
wire or anything. What's going to happen? Well, if I just have a
stationary charge sitting in a magnetic field, nothing really
is going to happen, right? Because we know that the force
due to a magnetic field is equal to the charge times the
cross product of the velocity of the charge and the
magnetic field. If this wires are stationary and
there's no voltage across it, et cetera, et cetera, the
velocity of this charge is going to be 0. So if the velocity is 0, we know
that the magnitude of a cross product is the
same thing as--. So Q is, that's just a scalar
quantity-- so that's just Q-- times the magnitude of the
velocity times the magnitude of B times sine theta. And in these situations where
anything that's going on in this plane is going to
be perpendicular to this magnetic field--. So the angle between the
magnetic field and any velocity-- if there were any
within this plane-- would be 90 degrees. So you wouldn't have to worry
about the sine theta too much. But we see if the velocity is
0 or the speed is 0, the magnitude of the velocity is 0,
that there's not going to be any net force due to the
magnetic field on this charge. And nothing interesting's
going to happen. But let's do a little
experiment. What happens if I were to move
this wire, if I were to shift it to the left, with
the velocity V? So I take this wire and I shift
it to the left with the velocity V. All right, so the whole wire
is shifting to the left. Well, if the whole wire is
shifting to the left, this charge is sitting on that wire,
so that charge is also going to move to the left
with the velocity V. And now things get
interesting. The charge is moving to the left
with the velocity V, so now we can apply the
first magnetism formula that we learned. We could apply this formula. So what's going to happen
to this charge? Well, the force of the charge
is going to be the charge times the magnitude of the
velocity cross the magnetic field vector. So we know that there's going
to be some net force. This is non-zero now. And this is non-zero,
we're assuming. And we're assuming the
charge is non-zero. So what direction is the
force going to be in? So let's do our right hand rule
on the cross product. V cross B will give
us the direction. So point your index finger in
the direction of the velocity. I have to look at my own
hand to make sure I'm doing it right. So you point your index
finger in the direction of the velocity. Point your middle finger
in the direction of the magnetic field. The magnetic field is popping
out of the page, so your middle finger is actually
going to be popping out of the page. Your next two fingers
are just going to do something like that. So you're kind of approximating
like you're shooting a gun. And then what's your
thumb going to do? Your thumb is going to
point straight up. This is the palm of--
that's your thumb. This could be your nail,
fingernail, fingernail of your thumb, fingernail of
your middle finger. This is the direction
of the velocity. Let me get a suitable color. The velocity is that way. The magnetic field is popping
out of the page. So the force on the particle--
on this charged particle or on this charge-- due to the
magnetic field is going to go in the direction
of your thumb. So the direction of the force
is in this direction. So what's going to happen? There's going to be a
net force in this direction on the charge. And the charge is going to
move upwards, right? I mean, when you start having
a moving-- you could imagine also that you had multiple
charges, right? If you had multiple charges
here and you're moving the whole wire, all of those
charges are going to be moving upwards. And what is another way to call
a bunch of moving charges along a conductor? Well, it's a current, depending
on how much charge is moving per second. So at least in very qualitative
terms, you see that when you move a wire
through a magnetic field or when you move a magnetic field
past a wire, right? Because they're kind of the same
thing, it's all about the relative motion. But if you move a wire through
a magnetic field, it is actually going to induce
a current in the wire. It's going to induce the current
in the wire, and actually this is how electric
generators are generated. And I'll do a whole series of
videos on how you-- you know, if you're using coal or steam or
hydropower, how that turns, essentially that turns these
generators around and it induces current. And that's how we get
electricity from all of these various energy sources
that essentially just make turbines turn. But anyway, let's go back
to what we were doing. So let me ask you a question. If this particle-- and this
all has a point-- if this particle starts at
the beginning. Let's say the particle
is right here. So it starts right where the
magnetic field starts affecting the wire. And how much work is going to be
done on the particle by the magnetic field? Well, what's work? Work is equal to force times
distance, where the force has to be in the same direction
as the distance, right? Force times distance, I
won't mess with the vectors right now. But they have to be in
the same direction. So how much work is going to
be done on this particle? So the work is going to be the
net force exerted on the particle times the distance. Well, this distance
is L, right? We say, once a particle gets
here there's no magnetic field up here, so the magnetic field
will stop acting on it. So the total work done: Work,
which is equal to force times distance, is equal to-- so the
net force is this up here. Q-- and I'll leave
some space-- V cross B times the distance. And the distance right here is
just a scalar quantity, so we could put it out front, right? Q times L times V
cross B, right? This is-- Q V cross B is the
force times the distance. That's just the work done. Now how much work is being
done per charge, right? This is how much work is being
done on this charge. But let's say there might have
been multiple charges, so we just want to know how much
work is done per charge. So work per charge. We could divide both
sides by charge. So work per charge is equal
to this per charge. So it is equal to the distance
times the velocity that you're pulling the wire to the left
with cross the magnetic field. This is where it gets
interesting. So what is work per charge? The units of work are
energy, right? Joules. And charge, that's
in coulombs. So what are joules
per coulomb? This is equal to volts. Volts are joules per coulomb. So this particle, or these
charges, are going to start moving in this direction as if
there is a voltage difference. As if there is a potential
difference between this point and this point. As if this is the positive
voltage terminal and this is the minus voltage terminal. So there's actually going to be
a voltage-- or a perceived voltage-- difference between
this point and this point that will start making the
current flow. Let's say you didn't even know
that there was a magnetic field here. You would just see this
current flowing. You'd be like, oh well, there
has to be a voltage difference there, right? But when we're dealing with
this-- because when we talk about voltages, that was like
a potential difference. That something-- that a particle
or a charge has a higher potential energy and
that's why it's moving. But it's hard to-- at least for
these purposes-- say, well you have a higher potential
energy here. It's really being created
by the magnetic field. So in this context, people
have said that instead of saying that this is creating a
voltage difference between this point and this point, if
the magnetic field on the moving wire is causing that,
people say that it's creating an electromotive force,
or an EMF. But EMF, the units are still
joules per coulomb or volts. And it really is-- in every way
when you're analyzing the circuits-- still the same
thing as a potential difference or as a voltage
difference. But since it seems a little bit
more proactive, it seems like this magnetic field is
actually impacting a force on this wire that is causing
the current to move. We call it EMF. So we could say that the EMF,
the electromotive force-- or the voltage across from here
to here, but they're really the same thing-- is equal to
the distance of the wire that's in the magnetic field
times the velocity-- that you're pulling the wire in--
cross the magnetic field. So let's say, I don't know,
let's just throw out a bunch of numbers. Let's say that the magnetic
field is-- I'll make it easy-- 2 teslas. My velocity to the left is
3 meters per second. And let's-- just for fun-- let's
give this a little bit of a resistance, just so we
can figure out something. So let's say this resistance is,
I don't know, let's say it is 6 ohms. There's a 6
ohm resistor here. So the resistance of the wire
from here to here is 6 ohms. All wires have some
resistance. So first of all,
what's the EMF? Oh, and let's say that
this total distance right here is 12 meters. So the EMF induced on the-- or
the electromotive force-- put on to the wire by the magnetic
field is going to equal the distance of the wire
in the magnetic field-- 12 meters-- times--. Well, when we're just taking
the cross product, we know that the velocity is
perpendicular to the magnetic field. So we don't have to worry about
sine theta because theta is already 90 degrees. So we just have to worry
about the magnitudes. So it's going to be 12 meters
times the velocity, which is 3 meters per second, times the
magnetic field, or the magnitude of the magnetic
field, that's 2 teslas. And so the EMF is 12
times 3 times 2. 12 times 6. Which is 72. You could say 72 volts, or
72 joules per coulomb. And now you have that potential
difference, or that EMF, across a 6 ohm
resistor, right? So that, you just go back to
voltage is equal to IR. Or you could write EMF
is equal to IR. So EMF divided by resistance. So if we take this EMF and we
divide it by the resistance-- divided by 6 ohms-- we get
the current, right? EMF divided by resistance
is equal to current. So you divide 72 volts or 72
joules per coulomb divided by 6 ohms. And then you get a
current going along this wire right here, due to the EMF, due
to the magnetic field-- I know it's very messy at this
point-- of 12 amperes. Anyway, I'm all out of time. I'll see you in the
next video.